Séminaire Lotharingien de Combinatoire, 78B.36 (2017), 12 pp.

Henri Mühle and Philippe Nadeau

The Alternating Group Generated by 3-Cycles

Abstract. We investigate the partial order on the alternating group generated by all 3-cycles. We first describe the cover relations in this poset. Permutations with odd cycles occur naturally, and we study the lower intervals they induce. These intervals are naturally embedded in the lattices of noncrossing partitions, and we provide several enumeration formulas for them. We also study the natural action of the braid group on the maximal chains in any given interval, and determine when this action is transitive. We also outline the many ways in which our construction can, or could, be extended.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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